Compute a · (b ✕ c), where a = i − 3j + k, b = 4i + j + k and c = 9i − j + 3k.

[tex]\mathbf a=\mathbf i-3\,\mathbf j+\mathbf k[/tex]
[tex]\mathbf b=4\,\mathbf i+\mathbf j+\mathbf k[/tex]
[tex]\mathbf c=9\,\mathbf i-\mathbf j+3\,\mathbf k[/tex]

[tex]\mathbf b\times\mathbf c=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\4&1&1\\9&-1&3\end{vmatrix}=4\,\mathbf i-3\,\mathbf j-13\,\mathbf k[/tex]

[tex]\mathbf a\cdot(\mathbf b\times\mathbf c)=(\mathbf i-3\,\mathbf j+\mathbf k)\cdot(2\,\mathbf i-3\,\mathbf j-13\,\mathbf k)=4+(-3)^2-13=0[/tex]